Question

How do you determine if $-10,20,-40,80$ is an arithmetic or geometric sequence?

Answer 1

Hint: In this question, we have to find whether the given sequence is arithmetic or geometric. Thus, we will use the progression rule to get the solution. As we know, an arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is equal to the difference between any other consecutive terms. Also, in geometric sequence, each term is multiplied by the previous term with the same constant. Thus, in this problem, we will find the common difference and the common ratio, to get whether the given sequence is arithmetic or the geometric sequence.

Complete step by step answer:
According to the question, we have to find whether the given sequence is arithmetic or the geometric sequence.
The sequence given to us is $-10,20,-40,80$ --------- (1)
Thus, we see from equation (1) that the first term is -10, second term is 20, and so on. Also, total number of terms is 4, that is
${{a}_{1}}=-10,{{a}_{2}}=20,{{a}_{3}}=-40,\text{ and }{{a}_{4}}=80$ -------- (2)
$n=4$
Now, we know that in arithmetic sequence, the difference between any two consecutive term are equal to each other, that is
$\Rightarrow d={{a}_{2}}-{{a}_{1}}={{a}_{3}}-{{a}_{2}}$
So, we will substitute the value of equation (2) in above equation, we get
$\Rightarrow d=20-\left( -10 \right)=-40-20$
On further solving the above equation, we get
$\Rightarrow d=20+10=-60$
Therefore, we get
$\Rightarrow d=30\ne -60$
Therefore, we see from the above equation that the difference between two consecutive terms is not each to other.
Thus, the given sequence is not an arithmetic sequence.
Now, we know that in geometric sequence, the ratio between any two consecutive term are equal to each other, that is
\[\Rightarrow r=\dfrac{{{a}_{2}}}{{{a}_{1}}}=\dfrac{{{a}_{3}}}{{{a}_{2}}}\]
So, we will substitute the value of equation (2) in above equation, we get
\[\Rightarrow r=\dfrac{20}{-10}=\dfrac{-40}{20}\]
On further solving the above equation, we get
\[\Rightarrow r=-2=-2\]
Therefore, we see from the above equation that the ratio between two consecutive terms is each to other.
Thus, the given sequence is the geometric sequence.
Therefore, the sequence $-10,20,-40,80$ is the geometric sequence.

Note:
While solving this problem, do mention the definition and formula of both the sequence in the solution to avoid confusion. For an accurate answer always put the numerator equal to the next term and the denominator as the previous term.